Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1212 Unicode version

Theorem bnj1212 28889
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1212.1  |-  B  =  { x  e.  A  |  ph }
bnj1212.2  |-  ( th  <->  ( ch  /\  x  e.  B  /\  ta )
)
Assertion
Ref Expression
bnj1212  |-  ( th 
->  x  e.  A
)
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ch( x)    th( x)    ta( x)    B( x)

Proof of Theorem bnj1212
StepHypRef Expression
1 bnj1212.1 . . 3  |-  B  =  { x  e.  A  |  ph }
21bnj21 28800 . 2  |-  B  C_  A
3 bnj1212.2 . . 3  |-  ( th  <->  ( ch  /\  x  e.  B  /\  ta )
)
43simp2bi 973 . 2  |-  ( th 
->  x  e.  B
)
52, 4bnj1213 28888 1  |-  ( th 
->  x  e.  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1649    e. wcel 1721   {crab 2678
This theorem is referenced by:  bnj1204  29099  bnj1296  29108  bnj1415  29125  bnj1421  29129  bnj1442  29136  bnj1452  29139  bnj1489  29143
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-rab 2683  df-in 3295  df-ss 3302
  Copyright terms: Public domain W3C validator